#8755448

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,25)(19,24,27,32)(20,22,30,31)(21,29,28,26)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(2,7)(5,8)(9,12)(13,15)(17,19,23,27)(18,24,25,32)(20,28,30,21)(22,26,31,29)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(2,7)(3,10)(9,12)(11,16)(18,25)(19,27)(20,30)(26,29)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20)(18,22)(19,28)(21,27)(23,30)(24,26)(25,31)(29,32)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,28)(18,26,25,29)(19,20,27,30)(22,24,31,32)(33,37,39,46)(34,41,36,44)(35,42,43,48)(38,45,40,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,22)(18,20)(19,29)(21,24)(23,31)(25,30)(26,27)(28,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662644);